MATH SOLVE

3 months ago

Q:
# Use logical equivalences (not a truth table) to reduce p → (q − p) to a tautology t. In other words, you should transform p → (q − p) into an equivalent statement, then transform that into another equivalent statement, and so on, until you arrive at a tautology. Your solution should look something like this: p → (q − p) = statement = statement = . . . . ... = t. (Note: the symbol = is technically the same as H. It's just easier to use = , because can easily be confused as being part of the logical statement you're transforming).

Accepted Solution

A:

Answer:The statement [tex]p\rightarrow (q\rightarrow p)[/tex] is a tautology.[tex]p\rightarrow (q\rightarrow p)\equiv \lnot p\lor (q\rightarrow p) \equiv \lnot p\lor (\lnot q\lor p) \equiv (p\lor \lnot p) \lor \lnot q \equiv T \lor \lnot q \equiv T[/tex]Step-by-step explanation:We have the following statement:[tex]p\rightarrow (q\rightarrow p)[/tex]To reduce the statement to a tautology we need to use the table of logical equivalences as follows:[tex]p\rightarrow (q\rightarrow p)\equiv[/tex][tex]\equiv \lnot p\lor (q\rightarrow p)[/tex] by the the logical equivalence involving conditional statement.[tex]\equiv \lnot p\lor (\lnot q\lor p)[/tex] by the the logical equivalence involving conditional statement.[tex]\equiv (p\lor \lnot p) \lor \lnot q[/tex] by the Associative law.[tex]\equiv T \lor \lnot q[/tex] by the Negation law.[tex]\equiv T[/tex] by the Domination law.